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Been playing Arena in MTG and it seems to allow casting of spells after resolution on the stack. This does not seem to follow the rules as written. Which is right.
For example three spells are on the stack and both people pass. the first one resolves and you are allowed to continue casting and add to the stack even though you had already begun resolving.
Allowing casting after resolution is according to rules as written. The relevant rules are 117.1a for when a player can cast a spell, and 117.3b which states that the active player get priority after spell resolution.
117.1a A player may cast an instant spell any time they have priority. A player may cast a noninstant spell during their main phase any time they have priority and the stack is empty.
117.3b The active player receives priority after a spell or ability (other than a mana ability) resolves.
So all players can cast add instants to the stack after the resolution of any spell or ability, although no non-instant can be cast until the stack is empty.
Yes, you can always cast spells after something on the stack resolves.
It seems as though your interpretation is that players add to the stack until they're done, and then the entire stack resolves from top to bottom until it's empty in one uninterruptible procedure. But that's not how it goes; there is no unstoppable stack-resolves-until-it's-empty mode we enter.
Instead, we just opt to let the top spell or ability on the stack resolve (via each player passing in succession). When it does resolve, the active player gets priority again, and can either add a new spell or ability to the stack, or can pass it to let others add something new to the stack.
This is actually integral to some abilities functioning properly: if you put two Shocks on the stack targeting my Snapping Sailback, requiring the stack to resolve as a whole would see my Snapping Sailback killed (it dies for having 4 toughness and 4 damage marked on it). But instead it works out like this:
